Modeling Multiplication

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3-4Modeling Multiplication 3-4

In this section students model an example such as 3 45 by showing 45on 3 Place mats and then combining the blocks in each place. Throughthis model, students intuitively recognize the distributive property of multi-

plication; that is, they see that 3 45 is the same as (3 4 tens) + (3 5).

This method is easily extended to multiplying a three-digit number by a one-

digit number. The process is the same and simply includes blocks-of-100. If

appropriate for your students, include three-digit numbers in these activities.

The basic goal of this section is for students to be able to model a multipli-

cation example or story problem with a single-digit multiplier. Not until the

next section will students predict outcomes and begin to develop mental

computation and paper-and-pencil techniques for multiplying. Many of the

ideas in these two sections will develop gradually over an extended period

of time.

The next two sections suggest a way in which the Digi-Block materials can be

used to guide students discovery of a multiplication algorithm.You may

want to be more or less directive in how the activities unfold.

Multiplying Tens and Ones

Present a story problem such as the following:

Three children each have 23 blocks.

How many blocks do they have together?

For demonstration, find a place whereyou can lay out three Place mats, oneabove the other. Have single blocks,

blocks-of-10, and empty holders avail-able. Ask each of three volunteers toshow 23 on one of the mats. To empha-size the partial products, ask,

How many groups of 20 do you see? How

many groups of 3 do you see?

Then ask,

How many groups of 23 do you see?

Focus Using base ten representations to model

multiplication with single-digit multipliers

You can use several Place mats to model multiplication oftwo-digit numbers. Shown here is the model for 3 23.


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To model multiplication as repeated addition, ask students to push together allthe blocks in each column, tell the total number of blocks-of-10 and ones (theorder doesnt matter), set the Digit Flip Cards to show the product, and writethe related multiplication sentence (3 23 = 69).

Have students follow a similar procedure for two or three more examples thatdo not require regrouping, such as 4 21, 3 32, and 2 42. For the lastexample, show students how they can work with one mat, placing one groupabove the other. This will save space when students are working independentlyor in small groups.

Whether using several mats or a single mat, make sure students can identifythe separate groups. Seeing the image of the repeated addition model rein-

forces the idea of multiplying the whole number, not

separate digits. Because of the limited space on Placemats, it is best to make no more than about 4 groups atthis stage. Once students are more familiar with theprocess, they can allow their groups to spill over the matand still know what numbers they are modeling.

Next present an example that requires regrouping, askingthree students to model 3 28. As with the previousexamples, students form the three separate groups on themat and then find the total number of blocks in eachplace. Note that students might count the blocks, add, or

recall a basic fact. Over time, encourage them to explaintheir approaches and discuss which is easiest.

This time, students must pack the blocks in order toset the Digit Flip Cards correctly. This should not bedifficult because it is the same process they have usedthroughout these units. You may want students to record,at the top of the mat, the number of blocks in each placebefore they pack. These subtotals help with later transla-tion to paper-and-pencil recording.

Continue with other examples with regrouping. Have students work in pairsor small groups to find 4 17 and 3 46. Provide time for students to recordtheir work. They can use drawings, words, and numbers to depict their steps.

The recording process provides another way for students to make multiplica-tion meaningful and is a vital component of their learning. Invite students toshare their recordings with one another, perhaps in relation to particular storyproblems. Over time, such work gives you important insight into studentsgrowth of understanding. At this stage, however, the focus is on performing

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For 3 28, students might record the numberin each place before packing.This is similar to

the intermediate steps of written computa-tions,when students record partial products.


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3-43-4the tasks with the blocks and communicating theprocess. More efficient recording schemes areexplored in the next section.

Students should have many opportunities to com-bine equal groups consisting of blocks-of-10 andsingle blocks. Connect the process to multiplicationexamples presented in both vertical and horizontalforms, as well as to a variety of story problems.Encourage students to ref lect on the process withquestions like these:

How many single blocks will there be when you put

them together?

How many new blocks-of-10 will there be? Why?

What does the number you are writing on the Place mattell you?

Working with Larger Numbers

Once again, the fact that all of the blocks look alikeexcept for size makes it easy for students to extendtheir thinking. When you decide students are ready,have them investigate multiplying one-digit by three-

digit numbers with an example such as 4 126. Theprocess is exactly the same as for two-digit numbers;students merely work with blocks-of-100 as well,applying their knowledge of packing the blocks.While you will want to consider a few more examples,avoid any that will require the use of several blocks-of-1000. Students who can model multiplication with two-digit numbers will quickly be able to do so with three-digit numbers.

Recording their work gives students achance to reflect on the process.

Students model 4 126 on a Place mat.

Then they pack and set the Digit Flip Cards to

show the product.

After combining the blocks,they write the total ineach place.


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Practicing Key Ideas

Roll a Product

Students work in pairs. One student forms a number on the Place mat.The other

student rolls a die.The number rolled tells how many groups to make of that numberon the mat. Together, students make the groups, find the product, and record the

corresponding multiplication sentence.

Assessing Learning

1. Have the student make 5 groups of 23 on the Place mat and set the DigitFlip Cards to show the product. Does the student make the groups correctly? find the correct answer?

2. Present 4 35 in vertical or horizontal form. Ask the student to show youhow to use the blocks to find the product and explain what he or she isthinking. Does the student model the example correctly? find the correct answer? clearly explain his or her thinking?

3. Have the student use the blocks to solve a story problem. For example:

The children made 3 buildings with blocks.

Each building has 48 blocks.

How many blocks are there in all?Does the student model the problem correctly? find the correct answer? clearly explain his or her thinking?

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Modeling Multiplication